Analysis for diffusion processes on Riemannian manifolds:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Singapore u.a.
World Scientific
2014
|
| Series: | Advanced series on statistical science & applied probability
18 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 379 p |
| ISBN: | 9789814452649 |
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Record in the Search Index
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| adam_text | Titel: Analysis for diffusion processes on Riemannian manifolds
Autor: Wang, Feng-Yu
Jahr: 2014
Contents
Preface v
1. Preliminaries 1
1.1 Riemannian manifold..................... 1
1.1.1 Differentiable manifold............... 1
1.1.2 Riemannian manifold................ 3
1.1.3 Some formulae and comparison results ...... 9
1.2 Riemannian manifold with boundary............ 11
1.3 Coupling and applications.................. 15
1.3.1 Transport problem and Wasserstein distance ... 16
1.3.2 Optimal coupling and optimal map........ 18
1.3.3 Coupling for stochastic processes.......... 19
1.3.4 Coupling by change of measure........... 22
1.4 Harnack inequalities and applications ........... 24
1.4.1 Harnack inequality ................. 24
1.4.2 Shift Harnack inequality.............. 31
1.5 Harnack inequality and derivative estimate........ 33
1.5.1 Harnack inequality and entropy-gradient estimate 33
1.5.2 Harnack inequality and i2-gradient estimate ... 36
1.5.3 Harnack inequalities and gradient-gradient estimates 37
1.6 Functional inequalities and applications.......... 39
1.6.1 Poincare type inequality and essential spectrum . 39
1.6.2 Exponential decay in the tail norm........ 42
1.6.3 The F-Sobolev inequality.............. 42
1.6.4 Weak Poincare inequality.............. 43
1.6.5 Equivalence of irreducibility and weak Poincare in-
equality ........................ 45
x Analysis for Diffusion Processes on Riemannian Manifolds
2. Diffusion Processes on Riemannian Manifolds without Boundary 49
2.1 Brownian motion with drift................. 49
2.2 Formulae for VPt and Ricz................. 54
2.3 Equivalent semigroup inequalities for curvature lower bound 60
2.4 Applications of equivalent semigroup inequalities..... 72
2.5 Transportation-cost inequality................ 77
2.5.1 From super Poincare to weighted log-Sobolev in-
equalities ....................... 79
2.5.2 From log-Sobolev to transportation-cost inequalities 82
2.5.3 From super Poincare to transportation-cost in-
equalities ....................... 87
2.5.4 Super Poincare inequality by perturbations .... 92
2.6 Log-Sobolev inequality: Different roles of Ric and Hess . . 95
2.6.1 Exponential estimate and concentration of/z ... 96
2.6.2 Harnack inequality and the log-Sobolev inequality 98
2.6.3 Hypercontractivity and ultracontractivity..... 102
2.7 Curvature-dimension condition and applications...... 109
2.7.1 Gradient and Harnack inequalities......... 109
2.7.2 HWI inequalities................... 120
2.8 Intrinsic ultracontractivity on non-compact manifolds . . 127
2.8.1 The intrinsic super Poincare inequality...... 129
2.8.2 Curvature conditions for intrinsic ultracontractivity 131
2.8.3 Some examples.................... 136
3. Reflecting Diffusion Processes on Manifolds with Boundary 141
3.1 Kolmogorov equations and the Neumann problem .... 142
3.2 Formulae for VPt,Ricz and I................ 146
3.2.1 Formula for VPt................... 146
3.2.2 Formulae for Ricz and I.............. 149
3.2.3 Gradient estimates ................. 152
3.3 Equivalent semigroup inequalities for curvature condition
and lower bound of I..................... 159
3.3.1 Equivalent Statements for lower bounds of Ric^
andl......................... 159
3.3.2 Equivalent inequalities for curvature-dimension
condition and lower bound of I........... 165
3.4 Harnack inequalities for SDEs on Rd and extension to non-
convex manifolds....................... 167
Contents xj
3.4.1 Construction of the coupling............ 169
3.4.2 Harnack inequality on Rd.............. 174
3.4.3 Extension to manifolds with convex boundary . . 176
3.4.4 Neumann semigroup on non-convex manifolds . . 180
3.5 Functional inequalities.................... 181
3.5.1 Estimates for inequality constants on compact
manifolds....................... 181
3.5.2 A counterexample for Bakry-Emery criterion . . . 184
3.5.3 Log-Sobolev inequality on locally concave
manifolds....................... 186
3.5.4 Log-Sobolev inequality on non-convex manifolds . 190
3.6 Modified curvature tensors and applications........ 195
3.6.1 Equivalent semigroup inequalities for the modified
curvature lower bound ............... 196
3.6.2 Applications of Theorem 3.6.1........... 200
3.7 Generalized maximum principle and Li-Yau s Harnack in-
equality ............................ 204
3.7.1 A generalized maximum principle......... 206
3.7.2 Li-Yau type gradient estimate and Harnack in-
equality ........................ 210
3.8 Robin semigroup and applications ............. 214
3.8.1 Characterization of P? w and V{€q)....... 215
3.8.2 Some criteria on XQ for fi(M) = 1......... 218
3.8.3 Application to HWI inequality........... 222
4. Stochastic Analysis on Path Space over Manifolds with Boundary 227
4.1 Multipücative functional................... 228
4.2 Damped gradient, quasi-invariant flows and Integration by
parts.............................. 234
4.2.1 Damped gradient Operator and quasi-invariant
flows......................... 234
4.2.2 Integration by parts formula............ 236
4.3 The log-Sobolev inequality.................. 239
4.3.1 Log-Sobolev inequality on W£........... 239
4.3.2 Log-Sobolev inequality on the free path space . . 242
4.4 Transportation-cost inequalities on path spaces over con-
vex manifolds......................... 245
4.5 Transportation-cost inequality on the path space over non-
convex manifolds....................... 251
xii Analysis for Diffusion Processes on Riemannian Manifolds
4.5.1 The case with a diffusion coefficient........ 252
4.5.2 Non-convex manifolds................ 255
5. Subelliptic Diffusion Processes 257
5.1 Functional inequalities.................... 259
5.1.1 Super and weak Poincare inequalities....... 259
5.1.2 Nash and log-Sobolev inequalities......... 265
5.1.3 Gruschin type Operator............... 275
5.1.4 Kohn-Laplacian type Operator........... 277
5.2 Generalized curvature and applications........... 283
5.2.1 Derivative inequalities................ 285
5.2.2 Applications of Theorem 5.2.1........... 289
5.2.3 Examples....................... 293
5.2.4 An extension of Theorem 5.2.1........... 300
5.3 Stochastic Hamiltonian system: Coupling method..... 303
5.3.1 Derivative formulae................. 304
5.3.2 Gradient estimates ................. 309
5.3.3 Harnack inequality and applications........ 317
5.3.4 Integration by parts formula and shift Harnack in-
equality ........................ 322
5.4 Stochastic Hamiltonian system: Malliavin calculus .... 328
5.4.1 A general result................... 328
5.4.2 Explicit formula................... 332
5.4.3 Two specific cases.................. 338
5.5 Gruschin type semigroups.................. 343
5.5.1 Derivative formula.................. 343
5.5.2 Log-Harnack inequality............... 353
Bibliography 365
Index 377
|
| any_adam_object | 1 |
| author | Wang, Feng-Yu |
| author_GND | (DE-588)1045333670 |
| author_facet | Wang, Feng-Yu |
| author_role | aut |
| author_sort | Wang, Feng-Yu |
| author_variant | f y w fyw |
| building | Verbundindex |
| bvnumber | BV041414872 |
| classification_rvk | SK 820 |
| classification_tum | MAT 605f |
| ctrlnum | (OCoLC)867145859 (DE-599)BVBBV041414872 |
| discipline | Mathematik |
| format | Book |
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| indexdate | 2024-07-10T00:56:15Z |
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| isbn | 9789814452649 |
| language | English |
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| series | Advanced series on statistical science & applied probability |
| series2 | Advanced series on statistical science & applied probability |
| spelling | Wang, Feng-Yu Verfasser (DE-588)1045333670 aut Analysis for diffusion processes on Riemannian manifolds Feng-Yu Wang Singapore u.a. World Scientific 2014 XII, 379 p txt rdacontent n rdamedia nc rdacarrier Advanced series on statistical science & applied probability 18 Chen, Jiageng 1874-1961 (DE-588)118871919 gnd rswk-swf Stochastische Analysis (DE-588)4132272-1 gnd rswk-swf Wissenschaftstheorie (DE-588)4117665-0 gnd rswk-swf Naturwissenschaften (DE-588)4041421-8 gnd rswk-swf Kreationismus (DE-588)4226745-6 gnd rswk-swf Schöpfung (DE-588)4053163-6 gnd rswk-swf Stabilisierung (DE-588)4357300-9 gnd rswk-swf Intelligent Design (DE-588)4787084-9 gnd rswk-swf Philosophie (DE-588)4045791-6 gnd rswk-swf Notenbankpolitik (DE-588)4130528-0 gnd rswk-swf Kreditmarkt (DE-588)4073788-3 gnd rswk-swf Diffusionsprozess (DE-588)4274463-5 gnd rswk-swf Riemannscher Raum (DE-588)4128295-4 gnd rswk-swf (DE-588)1071861417 Konferenzschrift gnd-content Stochastische Analysis (DE-588)4132272-1 s Diffusionsprozess (DE-588)4274463-5 s Riemannscher Raum (DE-588)4128295-4 s DE-604 Kreditmarkt (DE-588)4073788-3 s Stabilisierung (DE-588)4357300-9 s Notenbankpolitik (DE-588)4130528-0 s Chen, Jiageng 1874-1961 (DE-588)118871919 p Intelligent Design (DE-588)4787084-9 s Kreationismus (DE-588)4226745-6 s Wissenschaftstheorie (DE-588)4117665-0 s Naturwissenschaften (DE-588)4041421-8 s Schöpfung (DE-588)4053163-6 s Philosophie (DE-588)4045791-6 s Advanced series on statistical science & applied probability 18 (DE-604)BV011932321 18 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026862091&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Wang, Feng-Yu Analysis for diffusion processes on Riemannian manifolds Advanced series on statistical science & applied probability Chen, Jiageng 1874-1961 (DE-588)118871919 gnd Stochastische Analysis (DE-588)4132272-1 gnd Wissenschaftstheorie (DE-588)4117665-0 gnd Naturwissenschaften (DE-588)4041421-8 gnd Kreationismus (DE-588)4226745-6 gnd Schöpfung (DE-588)4053163-6 gnd Stabilisierung (DE-588)4357300-9 gnd Intelligent Design (DE-588)4787084-9 gnd Philosophie (DE-588)4045791-6 gnd Notenbankpolitik (DE-588)4130528-0 gnd Kreditmarkt (DE-588)4073788-3 gnd Diffusionsprozess (DE-588)4274463-5 gnd Riemannscher Raum (DE-588)4128295-4 gnd |
| subject_GND | (DE-588)118871919 (DE-588)4132272-1 (DE-588)4117665-0 (DE-588)4041421-8 (DE-588)4226745-6 (DE-588)4053163-6 (DE-588)4357300-9 (DE-588)4787084-9 (DE-588)4045791-6 (DE-588)4130528-0 (DE-588)4073788-3 (DE-588)4274463-5 (DE-588)4128295-4 (DE-588)1071861417 |
| title | Analysis for diffusion processes on Riemannian manifolds |
| title_auth | Analysis for diffusion processes on Riemannian manifolds |
| title_exact_search | Analysis for diffusion processes on Riemannian manifolds |
| title_full | Analysis for diffusion processes on Riemannian manifolds Feng-Yu Wang |
| title_fullStr | Analysis for diffusion processes on Riemannian manifolds Feng-Yu Wang |
| title_full_unstemmed | Analysis for diffusion processes on Riemannian manifolds Feng-Yu Wang |
| title_short | Analysis for diffusion processes on Riemannian manifolds |
| title_sort | analysis for diffusion processes on riemannian manifolds |
| topic | Chen, Jiageng 1874-1961 (DE-588)118871919 gnd Stochastische Analysis (DE-588)4132272-1 gnd Wissenschaftstheorie (DE-588)4117665-0 gnd Naturwissenschaften (DE-588)4041421-8 gnd Kreationismus (DE-588)4226745-6 gnd Schöpfung (DE-588)4053163-6 gnd Stabilisierung (DE-588)4357300-9 gnd Intelligent Design (DE-588)4787084-9 gnd Philosophie (DE-588)4045791-6 gnd Notenbankpolitik (DE-588)4130528-0 gnd Kreditmarkt (DE-588)4073788-3 gnd Diffusionsprozess (DE-588)4274463-5 gnd Riemannscher Raum (DE-588)4128295-4 gnd |
| topic_facet | Chen, Jiageng 1874-1961 Stochastische Analysis Wissenschaftstheorie Naturwissenschaften Kreationismus Schöpfung Stabilisierung Intelligent Design Philosophie Notenbankpolitik Kreditmarkt Diffusionsprozess Riemannscher Raum Konferenzschrift |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026862091&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV011932321 |
| work_keys_str_mv | AT wangfengyu analysisfordiffusionprocessesonriemannianmanifolds |